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A comment on Fourier transform – A math tool used in optics, MP3s, JPEGs and more (2013). Parent comment

Re:Fourier transform – A math tool used in optics, MP3s, JPE... by joshuamorton on Tuesday April 11, 2017 @12:56PM (#14089952)
>Sine and cosine waves whose frequencies are multiples of a fundamental frequency, form an orthonormal basis w.r.t. the inner product given by integrating f(x)ĝ(x)dx over one period of the sine

This, by the way, was perhaps the most mindblowing thing I learned in my advanced linear algebra course. I had already learned about applied Fourier Transforms and the FFT, but seeing the same tools we had used to prove things about Cartesian bases could be applied to something so unexpected, with such powerful results, was incredible.

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Re:Fourier transform – A math tool used in optics, MP3s, JPE... by blt on Tuesday April 11, 2017 @02:36PM (#14091049)
The "magic" of the Fourier transform and DFT is entirely contained in this proof, IMO. I think a lot of people find the Fourier transform extra-amazing because it is their first encounter with basis change and infinite dimensional vector spaces.

The FFT is still magic though :)

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